1. In winter weather, to frame the best compositions look for single objects that contrast with the scene's otherwise monochromatic look. If this is hard to come by, look for man-made objects that fit into the scene and contribute to the warm feeling of the image. On dreary days, I would recommend removing the sky from your framing, unless the sky is required to tell the story.
2. As I mentioned in my tips to making your landscape shots special, blue hour is a great time to photograph. Blue hour is a short period of time after sunset where the sky is reliably blue (no matter what the day's weather was). The lighting during that time makes it easy to shoot decorations and displays, and furthermore helps to add a warm mystique to your image. If you're a morning person, winter photographs are just as spectacular in the morning. I suggest waking up early for the unique reflections and colors a sunrise has to offer.
3. If you're trying to capture the snow actually falling, shoot at a slow exposure to capture the motion, or simply flash the scene to freeze the moment and capture the crystals of the snow.
4. Include people! The cold winter weather means that people of all ages grab their colorful jacket that has been sitting in their closet for the past year. Use this to your advantage, by creating a contrast between the white snow and the red, yellow, blue, green and orange clothes.
5. Remember that snow very easily tricks your automatic white balance and ISO settings. Make sure to look at your image histogram (if possible) or your image preview, and try to compensate for all errors through manual settings.
Now go out, be safe and enjoy the holiday season.